Topological Derivative for Three-Dimensional Linear Elasticity Problems

Abstract

In this chapter the elasticity boundary value problems in three spatial dimensions are considered. The topological derivative of the total potential energy for the perturbation in the form of a small spherical cavity $B_{\varepsilon}(\widehat{x})$, with $\widehat{x} \in \Omega \subset \mathbb{R}^{3}$ and $\overline{B_{\varepsilon}} \Subset \Omega$, is obtained. Therefore, the topologically perturbed domain is given by $\Omega_{\varepsilon} = \Omega \setminus \overline{B_{\varepsilon}}$.